Question: For numbers $16$ to $17$:
In the figure, segment $SU$ and $WU$ are secant segments while segment $XU$
is a tangent segment. Given that $WU = 14$, $ST = 12$, $TU = 4$ and
$mSW = 160^\circ$ answer the following:
Graph/shapes: A circle is shown in the center-right with points $S$ at the top, $X$ on the left side of the circle, $W$ on the lower-right side, $V$ near the lower part of the circle, $T$ on the lower-left interior chord, and $U$ at the bottom-left outside the circle. A tangent segment $XU$ extends from $U$ to the circle at $X$. Two secant segments from $U$ pass through the circle: $U-T-S$ and $U-V-W$. Labels $12$ and $4$ appear along the secant $U-S$, and $14$ is written along the lower exterior segment toward $W$. position_hint = center
If $mTV = 60^\circ$, what is the measure of angle $SUW$?
Select one:
a. $110^\circ$
b. $160^\circ$
c. $50^\circ$
d. $60^\circ$
Find the measure of segment $XU$.
Select one:
a. $8$
b. $2\sqrt{2}$
c. $4$
d. $4\sqrt{3}$
1. **State the problem:**
We have a circle with two secant segments $SU$ and $WU$ and a tangent segment $XU$ from point $U$ outside the circle.
Given:
- $WU = 14$
- $ST = 12$
- $TU = 4$
- $mSW = 160^\circ$
- $mTV = 60^\circ$
We need to find:
- The measure of angle $SUW$
- The length of segment $XU$
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2. **Find the measure of angle $SUW$: **
Angle $SUW$ is formed by two secants $SU$ and $WU$ intersecting outside the circle at $U$.
The measure of the angle formed by two secants intersecting outside a circle is half the difference of the intercepted arcs.
Formula:
$$m\angle SUW = \frac{1}{2} |m\overset{\frown}{SW} - m\overset{\frown}{TV}|$$
Given:
- $m\overset{\frown}{SW} = 160^\circ$
- $mTV = 60^\circ$ (angle at $T$ and $V$ on the circle, so arc $TV = 60^\circ$)
Calculate:
$$m\angle SUW = \frac{1}{2} |160^\circ - 60^\circ| = \frac{1}{2} \times 100^\circ = 50^\circ$$
So, the measure of angle $SUW$ is $50^\circ$.
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3. **Find the length of segment $XU$: **
Since $XU$ is a tangent segment and $SU$, $WU$ are secants from point $U$, the tangent-secant theorem applies:
The square of the length of the tangent segment equals the product of the lengths of the entire secant segment and its external part.
Formula:
$$XU^2 = WU \times UV$$
We know:
- $WU = 14$
- $ST = 12$
- $TU = 4$
Since $SU$ is a secant with parts $ST$ and $TU$, total $SU = ST + TU = 12 + 4 = 16$.
For $WU$, the external segment is $WU = 14$, and the internal segment $VW$ is unknown.
Using the secant-secant theorem:
$$SU \times TU = WU \times UV$$
Plug in known values:
$$16 \times 4 = 14 \times UV$$
$$64 = 14 \times UV$$
$$UV = \frac{64}{14} = \frac{32}{7}$$
Now apply tangent-secant theorem:
$$XU^2 = WU \times UV = 14 \times \frac{32}{7} = 2 \times 32 = 64$$
Therefore:
$$XU = \sqrt{64} = 8$$
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**Final answers:**
- $m\angle SUW = 50^\circ$
- $XU = 8$